Iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and a class of relations among multiple zeta values

TL;DR

This paper introduces confluence relations derived from iterated integrals on punctured projective lines, which potentially encompass all linear relations among multiple zeta values, including known shuffle and duality relations.

Contribution

It defines a new class of linear relations among multiple zeta values from iterated integrals, suggesting these relations may generate all existing relations.

Findings

Confluence relations imply regularized double shuffle relations.

Confluence relations imply duality relations.

Proposed relations potentially exhaust all linear relations among multiple zeta values.

Abstract

In this paper we consider iterated integrals on $\mathbb{P}^{1}\setminus\{0,1,\infty,z\}$ and define a class of $\mathbb{Q}$-linear relations among them, which arises from the differential structure of the iterated integrals with respect to $z$. We then define a new class of $\mathbb{Q}$-linear relations among the multiple zeta values by taking their limits of $z\rightarrow1$, which we call \emph{confluence relations} (i.e., the relations obtained by the confluence of two punctured points). One of the significance of the confluence relations is that it gives a rich family and seems to exhaust all the linear relations among the multiple zeta values. As a good reason for this, we show that confluence relations imply both the regularized double shuffle relations and the duality relations.